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---------

Co-authored-by: Jason Landini <[email protected]>

applications/CHAC_anisotropy/CHAC_anisotropy.md

@@ -1,4 +1,4 @@
-## PRISMS-PF: CHAC Anisotropy (with Coupled CH-AC Dynamics)
+# PRISMS-PF: CHAC Anisotropy (with Coupled CH-AC Dynamics)
 Consider a free energy expression of the form:
 
 $$

applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.md

@@ -1,4 +1,4 @@
-## PRISMS PhaseField: Regularized Anisotropy (with Coupled CH-AC Dynamics)
+# PRISMS PhaseField: Regularized Anisotropy (with Coupled CH-AC Dynamics)
 
 Consider a free energy expression of the form:
 

applications/CHAC_performance_test/formulation_coupledCHAC.md

@@ -1,4 +1,4 @@
-## PRISMS PhaseField: Coupled Cahn-Hilliard and Allen-Cahn Dynamics
+# PRISMS PhaseField: Coupled Cahn-Hilliard and Allen-Cahn Dynamics
 
 Consider a free energy expression of the form:
 

applications/allenCahn_conserved/allenCahn_conserved.md

@@ -0,0 +1,183 @@
+# PRISMS PhaseField: Globally Conserved Allen-Cahn Dynamics
+
+This application performs a  phase field simulation of Allen-Cahn dynamics subject to global (as opposed to local) conservation. Global conservation implies that $\int_\Omega \eta \,dV$, where $\Omega$ is the volume of the system, is constant in time.
+
+Note: This application needs to run with **uniform mesh** in order for the calculation of the chemical potential to be accurate.
+
+Consider a free energy expression of the form:
+
+$$
+\begin{equation}
+  \Pi(\eta, \nabla  \eta) = \int_{\Omega}    f( \eta ) + \frac{\kappa}{2} \nabla  \eta  \cdot \nabla  \eta    ~dV, 
+\end{equation}
+$$
+
+where $\eta$ is the structural order parameter, and $\kappa$ is the gradient length scale parameter.
+
+## Variational treatment
+
+Considering variations on the primal field $\eta$ of the from $\eta+\epsilon w$, we have
+
+$$
+\begin{align}
+\delta \Pi &=  \left. \frac{d}{d\epsilon} \int_{\Omega}  f(\eta+\epsilon w) +  \frac{\kappa}{2} \nabla  (\eta+\epsilon w)  \cdot  ~\nabla  (\eta+\epsilon w)   ~dV \right\vert_{\epsilon=0}
+\end{align}
+$$
+
+$$
+\begin{align}
+&=  \int_{\Omega}   w f_{,\eta} +   \kappa \nabla w \nabla  \eta    ~dV
+\end{align}
+$$
+
+$$
+\begin{align}
+&=  \int_{\Omega}   w \left( f_{,\eta} -  \kappa \Delta \eta \right)  ~dV  +   \int_{\partial \Omega}   w \kappa \nabla \eta \cdot n   ~dS,
+\end{align}
+$$
+
+where $f_{,\eta} = \partial f/\partial \eta$.
+
+Assuming $\kappa \nabla \eta \cdot n = 0$, and using standard variational arguments on the equation $\delta \Pi =0$ we have the expression for chemical potential as
+
+$$
+\begin{equation}
+  \mu  = f_{,\eta} -  \kappa \Delta \eta.
+\end{equation}
+$$
+
+## Kinetics
+The Parabolic PDE for Allen-Cahn dynamics is given by:
+
+$$
+\begin{align}
+\frac{\partial \eta}{\partial t} &= -M\mu,
+\end{align}
+$$
+
+where $M$ is the constant mobility. However, the previous equation does not ensure global conservation of $\eta$. In order to achieve global conservation we can add a spatially-uniform term, $A$, to the RHS that effectively offsets the total change in $\eta$ such that $\int_\Omega \partial \eta / \partial t\,dV =0$:
+
+$$
+\begin{align}
+\frac{\partial \eta}{\partial t} &= -M\mu+A.
+\end{align}
+$$
+
+Applying the global conservation constraint to previous equation, we obtain
+
+$$
+\begin{align}
+\int_\Omega \frac{\partial \eta}{\partial t}\,dV &= -\int_\Omega (M\mu-A) \,dV = 0.
+\end{align}
+$$
+
+Since $A$ is spatially uniform and $M$ is a constant, $A$ given by
+
+$$
+\begin{align}
+A &= \frac{M}{V}\int_\Omega \mu \,dV.
+\end{align}
+$$
+
+Subsituting $A$ from
+
+$$
+\begin{align}
+A &= \frac{M}{V}\int_\Omega \mu \,dV.
+\end{align}
+$$
+
+into 
+
+$$
+\begin{align}
+\frac{\partial \eta}{\partial t} &= -M\mu+A.
+\end{align}
+$$
+
+we get
+
+$$
+\begin{align}
+\frac{\partial \eta}{\partial t} &= -M(\mu-\bar{\mu}),
+\end{align}
+$$
+
+where $\bar{\mu} = (1/V)\int_\Omega \mu \,dV$.
+
+## Time discretization
+Considering forward Euler explicit time stepping, we have the time discretized kinetics equations:
+
+$$
+\begin{align}
+\eta^{n+1} &= \eta^{n} - \Delta t M(\mu^n-\bar{\mu}^n)
+\end{align}
+$$
+
+and
+
+$$
+\begin{align}
+\mu^{n+1} &= f_{,\eta}^n -  \kappa \Delta \eta^n.
+\end{align}
+$$
+
+## Weak formulation
+In the weak formulation, considering an arbitrary variation $w$, the above equation can be expressed as a residual equations:
+
+$$
+\begin{align}
+\int_{\Omega}   w \eta^{n+1} ~dV&= \int_{\Omega}  w [ \eta^{n} - \Delta t M(\mu^n-\bar{\mu}^n) ]~dV 
+\end{align}
+$$
+
+$$
+\begin{align}
+r_{\eta} &= \eta^{n} - \Delta t M(\mu^n-\bar{\mu}^n)
+\end{align}
+$$
+
+and
+
+$$
+\begin{align}
+\int_{\Omega}   w \mu^{n+1} ~dV&= \int_{\Omega} [w f_{,\eta}^n - w \kappa \Delta \eta^{n}]~dV 
+\end{align}
+$$
+
+$$
+\begin{align}
+&= \int_{\Omega}   w (  f_{,\eta}^{n} )  + \nabla w \cdot (\kappa \nabla \eta^{n}) ~dV, 
+\end{align}
+$$
+
+$$
+\begin{align}
+r_{\mu x} &=  \kappa \nabla \eta^{n}
+\end{align}
+$$
+
+$$
+\begin{align}
+r_{\mu} &=  f_{,\eta}^{n} 
+\end{align}
+$$
+
+where the reference chemical potential $\bar{\mu}^n$ for time step $n$ in
+
+$$
+\begin{align}
+\int_{\Omega}   w \eta^{n+1} ~dV&= \int_{\Omega}  w [ \eta^{n} - \Delta t M(\mu^n-\bar{\mu}^n) ]~dV 
+\end{align}
+$$
+
+is calculated as
+
+$$
+\begin{align}
+\bar{\mu}^n& = \frac{1}{V}\int_\Omega \mu^n ~dV.
+\end{align}
+$$
+
+The above values of  $r_{\eta}$, $r_{\mu}$, and $r_{\mu x}$ are used to define the residuals in the following parameters file:
+`\textit{applications/allenCahn\_conserved/equations.cc`